A signal, coined PEGS for Prompt Elasto-Gravity Signal, was recently identified on seismograms preceding the seismic waves generated by very large earthquakes, opening promising applications for earthquake and tsunami early warning. Nevertheless, this signal is about 1,000,000 times smaller than seismic waves, making its use in operational warning systems very challenging. A Deep Learning algorithm, called PEGSNet, was later designed to estimate, as fast as possible, the magnitude of an ongoing large earthquake from PEGS recorded in real time. PEGSNet was applied to Japan and Chile and proved capable of tracking the magnitude of the M_w 9.1 Tohoku-oki and M_w 8.8 Maule earthquakes within a few minutes from the events origin times. Here, we apply this algorithm to a very well instrumented region: Alaska. We find that, applied to such a dense seismic network, the performance of PEGSNet is drastically improved, with robust performances obtained for earthquakes with magnitudes down to 7.8. The gain in resolution also allows us to estimate the focal mechanism of the events in real time, providing all the information required for tsunami warning within less than 3 minutes.